deletions | additions       diff --git a/sectionIntroduction_.tex b/sectionIntroduction_.tex  index 99cfa3d..1808e7f 100644  --- a/sectionIntroduction_.tex  +++ b/sectionIntroduction_.tex 
...
Matrices are one of the most fundamental objects in the mathematical description of nature, and as such they are ubiquitous in every area of science. For example, they arise naturally in all types of linear response theories as the first term in a multidimensional Taylor series, encoding the response of each component of the system to each component of the stimulus. Hence, in many scientific applications, matrices contain the essential information about the system being studied.  Despite their ubiquity, the The  calculation of matrices often requires considerable computational effort. Returning to the linear response theory example, it might be necessary to individually calculate the response of every component of the system to every component of the stimulus and, depending on the area of application, each individual computation may itself be quite expensive. The overall expense stems from the fact that evaluating a matrix of dimension \(N\times M\) requires, in principle, the individual evaluation of \(N\times M\) elements. But this does not always have to be the case. For example, if we know \emph{a priori} the eigenvectors of a \(N\times N\) diagonalizable matrix, then we can obtain the full matrix by only calculating the \(N\) diagonal elements. Similarly, a sparse matrix, which contains many zero elements, can be evaluated by calculating only the non-zero elements, if we know in advance where such elements are located.